Designs related through projective and Hopf maps

We formalize constructions which use the $\Bbb F$-projective map to build a spherical $t$-design on $S^d$ from a projective $\left\lfloor\frac t2\right\rfloor$-design on a real, complex, quaternionic, or octonionic projective space $\Bbb{FP}^n\neq \Bbb{OP}^2$ and spherical $t$-designs on $S^k$ or to build a projective $\left\lfloor\frac t2\right\rfloor$-design on $\Bbb{FP}^n$ from a spherical $t$-design on $S^d$, where $k=(\Bbb F\::\:\Bbb R)-1$ and $d=(k+1)n+k$. We prove that the $n=1$ cases of these constructions give rise to analogous constructions which use generalized Hopf maps to relate spherical $t$-designs on $S^{2k+1}$ to spherical $\left\lfloor\frac t2\right\rfloor$-designs on $S^{k+1}$. This generalizes work of K\"{o}nig and Kuperberg, who proved validity of the $\Bbb F=\Bbb C$ case of the projective constructions, and of Okuda, who proved validity of the $\Bbb F=\Bbb C$ case of the Hopf constructions.Comment: 20 pages, 5 figures. New related references cited, class changed to amsart, minor typos fixe

Optimal measures for p-frame energies on spheres

We provide new answers about the distribution of mass on spheres so as to minimize energies of pairwise interactions. We find optimal measures for the pp-frame energies, i.e., energies with the kernel given by the absolute value of the inner product raised to a positive power pp. Application of linear programming methods in the setting of projective spaces allows for describing the minimizing measures in full in several cases: we show optimality of tight designs and of the 600-cell for several ranges of pp in different dimensions. Our methods apply to a much broader class of potential functions, namely, those which are absolutely monotonic up to a particular order

Octonions and the two strictly projective tight 5-designs

In addition to the vertices of the regular hexagon and icosahedron, there are precisely two strictly projective tight 5-designs: one constructed from the short vectors of the Leech lattice and the other corresponding to a generalized hexagon structure in the octonion projective plane. This paper describes a new connection between these two strictly projective tight 5-designs -- a common construction using octonions. Certain octonion involutionary matrices act on a three-dimensional octonion vector space to produce the first 5-design and these same matrices act on the octonion projective plane to produce the second 5-design. This result uses the octonion construction of the Leech lattice due to Robert Wilson and provides a new link between the generalized hexagon Gh(2,8) and the Leech lattice.Comment: Accepted version with significant revisions based on reviewer comments. New title and abstract. 12 page

Computer-assisted proofs in geometry and physics

Thesis (Ph. D.)--Massachusetts Institute of Technology, Department of Mathematics, 2013.Cataloged from PDF version of thesis.Includes bibliographical references.In this dissertation we apply computer-assisted proof techniques to two problems, one in discrete geometry and one in celestial mechanics. Our main tool is an effective inverse function theorem which shows that, in favorable conditions, the existence of an approximate solution to a system of equations implies the existence of an exact solution nearby. This allows us to leverage approximate computational techniques for finding solutions into rigorous computational techniques for proving the existence of solutions. Our first application is to tight codes in compact spaces, i.e., optimal codes whose optimality follows from linear programming bounds. In particular, we show the existence of many hitherto unknown tight regular simplices in quaternionic projective spaces and in the octonionic projective plane. We also consider regular simplices in real Grassmannians. The second application is to gravitational choreographies, i.e., periodic trajectories of point particles under Newtonian gravity such that all of the particles follow the same curve. Many numerical examples of choreographies, but few existence proofs, were previously known. We present a method for computer-assisted proof of existence and demonstrate its effectiveness by applying it to a wide-ranging set of choreographies.by Gregory T. Minton.Ph.D